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Semismooth Newton Method for Boundary Bilinear Control

Eduardo Casas, Konstantinos Chrysafinos, Mariano Mateos

Abstract

We study a control-constrained optimal control problem governed by a semilinear elliptic equation. The control acts in a bilinear way on the boundary, and can be interpreted as a heat transfer coefficient. A detailed study of the state equation is performed and differentiability properties of the control-to-state mapping are shown. First and second order optimality conditions are derived. Our main result is the proof of superlinear convergence of the semismooth Newton method to local solutions satisfying no-gap second order sufficient optimality conditions as well as a strict complementarity condition.

Semismooth Newton Method for Boundary Bilinear Control

Abstract

We study a control-constrained optimal control problem governed by a semilinear elliptic equation. The control acts in a bilinear way on the boundary, and can be interpreted as a heat transfer coefficient. A detailed study of the state equation is performed and differentiability properties of the control-to-state mapping are shown. First and second order optimality conditions are derived. Our main result is the proof of superlinear convergence of the semismooth Newton method to local solutions satisfying no-gap second order sufficient optimality conditions as well as a strict complementarity condition.
Paper Structure (6 sections, 24 theorems, 116 equations, 2 tables)

This paper contains 6 sections, 24 theorems, 116 equations, 2 tables.

Table of Contents

  1. Introduction
  2. State equation
  3. Analysis of the optimal control problem
  4. Convergence of the semismooth Newton method
  5. A numerical example and some computational considerations
  6. Proofs of some auxiliary results

Key Result

Theorem 2.3

There exists $\mu>0$ such that for every $u \in \mathcal{A}_0$ equation E01 has a unique solution $y_u \in Y:= H^1(\Omega) \cap C^{0,\mu}(\bar{\Omega})$. Furthermore, the following estimates hold: where $C$, $M_\infty$ and $C_{\mu,\infty}$ are independent of $u$.

Theorems & Definitions (46)

  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • ...and 36 more